An Effective Number Geometric Method of Computing the Fundamental Units of an Algebraic Number Field

The Minkowski method of unit search is applied to particular types of parallelotopes permitting to discover algebraic integers of bounded norm in a given algebraic number field of degree n at will by solving successively $2n$ linear inequalities for one unknown each. Application is made to the unit search for all totally real number fields of minimal discriminant for $n \leqslant 7$.